Function value at different points

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Discussion Overview

The discussion revolves around the possibility of determining the value of a function defined in three-dimensional space at arbitrary points based on its known values at a limited number of specific points. Participants explore concepts related to interpolation and function fitting in the context of both theoretical and practical applications.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants suggest that if a function is defined at a finite number of points, interpolation methods, such as Lagrange polynomials, can be used to estimate values at other points.
  • Others argue that knowing the function's value at a limited number of points does not uniquely determine the function elsewhere, as multiple functions can fit the same data points but diverge at other locations.
  • A participant highlights the importance of understanding the type of function being considered, noting that without this information, no finite set of points can uniquely define the function.
  • Some participants propose starting with simpler cases, such as one-dimensional functions, to illustrate the limitations of curve fitting and interpolation.

Areas of Agreement / Disagreement

Participants generally agree that knowing function values at a finite number of points does not guarantee a unique function elsewhere. However, there is no consensus on the implications of this for interpolation methods or the types of functions being discussed.

Contextual Notes

The discussion touches on various mathematical concepts, including interpolation and curve fitting, but does not resolve the complexities involved in defining functions based on limited data points.

Apashanka
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If for a field say ##f=xyz## ,it's value is defined at 10 points in the 3-D Cartesian co-ordinate system...now using these 10 values of f and the corresponding coordinates is it possible to find the value of f at any ##(x,y,z)## of choice??
 
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Do you mean 10 points in total, or 10 points along each axis (ie. 1000 points in total)?

Either you are trying to fit a function to given values at the grid points, in which case you to get values elsewhere you must interpolate using, for example, lagrange polynomials (MATLAB solution here), or your function is defined by an expression in closed form in which case you already know its value everywhere.
 
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pasmith said:
Do you mean 10 points in total, or 10 points along each axis (ie. 1000 points in total)?

Either you are trying to fit a function to given values at the grid points, in which case you to get values elsewhere you must interpolate using, for example, lagrange polynomials (MATLAB solution here), or your function is defined by an expression in closed form in which case you already know its value everywhere.
(10) points in total
 
It was the second paragraph of pasmith's that was most important.
 
It's ok if ##f(x,y,z)## is known for 5 values of ##(x,y,z)## then how will the formula of ##f(x,y,z)## at any ##x,y,z## change??for 1-D it is simple
 
Apashanka said:
If for a field say ##f=xyz## ,it's value is defined at 10 points in the 3-D Cartesian co-ordinate system...now using these 10 values of f and the corresponding coordinates is it possible to find the value of f at any ##(x,y,z)## of choice??

The value of ##f## at 10 points does not determine a unique value of ##f## at other points. You can invent many different functions that agree with ##f## on the 10 points but disagree with each other elsewhere. The general topic of inventing a function that has specified values on a finite number of points can be studied under the topics of "fitting a function to data" or "interpolation".

In your example, a straightforward method is to use multidimensional Lagrange interpolation. I think Answer 1 to https://math.stackexchange.com/ques...omial-in-3-d-variable-of-interest-is-a-vector describes this method.

The wikipedia gives a list of multivariate interpolation methods suited to different applications. https://en.wikipedia.org/wiki/Multivariate_interpolation
 
No. I would think about this in 1-D first. How many points does it take to determine a quadratic function ( f(x) = ax2 +bx +c )? Would those points define a unique cubic equation?
More info here: https://en.wikipedia.org/wiki/Curve_fitting
 
DaveE said:
How many points does it take to determine a quadratic function ( f(x) = ax2 +bx +c )? Would those points define a unique cubic equation?

The OP just said "function", not "quadratic function" or "cubic function". With no information about what kind of function it is, no finite set of points can tell you the function.
 
PeterDonis said:
The OP just said "function", not "quadratic function" or "cubic function". With no information about what kind of function it is, no finite set of points can tell you the function.
Yes, I saw that. That's why my first sentence was "No."
I was suggesting he look at a simpler problem to highlight the limitations of curve fitting. If you can't do it for simple polynomials (even the 11th order ones), it won't work in the more general case.
 
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